1. Field of Invention
The present invention relates generally to analyzing integrated circuits and, more particularly, to techniques for performing simulations of radio frequency (RF) integrated circuits.
2. Discussion of Background
The exploding demand for high performance wireless products has increased the need for efficient and accurate simulation techniques for RF integrated circuits. RF circuit simulation is difficult because RF circuits typically contain signals with multiple-timescale properties, as usually the data and carrier signals in a system are separated in frequency by several orders of magnitude. Special-purpose RF simulators exploit the sparsity of the spectrum in order to make the computations tractable. See K. S. Kundert, “Introduction to RF Simulation and Its Applications,” IEEE J. Sol. State Circuits, September 1999. For example, instead of performing a long transient analysis of a circuit driven by a periodic source, we may seek to find the steady-state directly. Unfortunately, periodic-steady-state computation may be problematic in that it can be inefficient and inaccurate using conventional methods.
Two numerical methods commonly used in steady-state computation are the shooting-Newton method, based on low-order finite difference discretizations such as the second-order Gear method (see K. S. Kundert, “Introduction to RF Simulation and Its Applications,” IEEE J. Sol. State Circuits, September 1999), and the harmonic balance method, based on high-order spectral discretizations (see “Nonlinear Circuit Analysis Using The Method of Harmonic Balance-A Review of The Art,” Part I-Introductory Concepts, Int. J. Microwave and Millimeter Wave Computer Aided Engineering, vol. 1, no. 1, 1991).
An advantage of the low-order polynomial-based methods is that these methods can select time-points based on localized error estimates and as a result can easily handle sharp transitions in circuit waveforms. The harmonic balance method, on the other hand, has the advantage of attaining spectral accuracy for smooth waveforms. Recent developments in matrix-free Krylov-subspace algorithms have made these methods even more popular as they can be used to analyze circuits with thousands of devices. For more information on matrix-free Kylov-subspace algorithms, see R. Telichevesky, K. Kundert and J. K. White, “Efficient AC and Noise Analysis of Two-Tone RF Circuits,” Proc. 33rd Design Automation Conference, June, 1996; D. Long and R. Melville and K. Ashby and B. Horton, “Full Chip Harmonic Balance,” Proc. Custom Integrated Circuits Conference, May, 1997.
High precision computations are often necessary in RF circuit simulation. For example, accurately computing the noise figure of a highly nonlinear RF circuit often requires accurate determination of the periodic operating point up to many multiples of the fundamental frequency, as noise may be translated from very high frequencies by the mixing action of the time-varying circuit elements to appear at the output.
High accuracy is easy to achieve for smooth waveforms using spectral approximation in the harmonic balance method. However, non-linear circuits often produce waveforms that have sharp-transition regions. The waveforms may only be C0 in the case of a pulse, which means that the waveforms are continuous but the derivatives of the waveforms are not continuous at some points. In this case, the spectral accuracy of the harmonic balance method is lost and acceptable accuracy is obtained only at the cost of including very many harmonics and/or timepoints in the simulation.
To help handle sharp transitions, Nastov and White introduced the time-mapped harmonic balance (TMHB) method. See O. J. Nastov and J. K. White, “Time-Mapped Harmonic Balance,” Proc. 36th Design Automation Conference, New Orleans, La., June, 1999. In the TMHB method, a time-map function is used to map the solution in the original time space to a space where the solution is more smooth. This method is an improvement over traditional harmonic balance, but is still restricted, since, theoretically speaking, a C0 point in the original time space is still a C0 point in the mapped time space, and so one still cannot achieve spectral accuracy. Basically, the TMHB method can only put more points around the C0 points to localize the oscillations inevitably associated with using a spectral method to approximate C0 functions. And, as with all methods that utilize global basis functions, for reasons of numerical stability there are also strong restrictions on how rapidly the timestep can be changed locally.
The low-order multi-step methods, such as the Gear methods, typically used in the shooting method offer excellent flexibility in locally adapting the timesteps. For low to moderate accuracy, these methods are preferred for problems that are highly nonlinear and/or contain sharp transitions. However, low-order multi-step methods are not efficient at high precision because very fine discretizations are needed, resulting in loss of speed and increased memory requirements. In addition, the multistep discretizations used in the shooting method must be causal, and so have stability problems associated with using one-side approximations, namely, performing differentiation using backward differences. It is generally accepted that multistep methods of order greater then three or four are not sufficiently stable, and in practice even the third order methods have relatively stringent stability restrictions on how rapidly timesteps may be varied. As a result, multi-step methods of order higher than 2 are not widely used in circuit simulation. Often, because of the backward-looking nature of the approximation, it is necessary to decrease the order after sharp-transition points or C0 points, often to first order, resulting in very fine grids in those regions.
In sum, most RF circuit analysis tools use either shooting-Newton or harmonic balance methods. Unfortunately, neither method can efficiently achieve high accuracy on strongly nonlinear circuits possessing waveforms with rapid transitions.